differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, the Kirwan map, introduced by British mathematician

Frances Kirwan Dame Frances Clare Kirwan, (born 21 August 1959) is a British mathematician, currently Savilian Professor of Geometry at the University of Oxford. Her fields of specialisation are algebraic and symplectic geometry. Education Kirwan was educ ...

, is the

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

H^*_G(M) \to H^*(M /\!/_p G)

where *

M

is a Hamiltonian G-space; i.e., a

symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...

acted by a Lie group ''G'' with a

moment map In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the ac ...

\mu: M \to ^*

. *

H^*_G(M)

is the

equivariant cohomology ring In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an or ...

M

; i.e.. the cohomology ring of the

homotopy quotient In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group , is a specific bundle over a classifying space , such that every bundle with the given structure group over is a pullback by mea ...

EG \times_G M

M

G

. *

M /\!/_p G = \mu^(p)/G

is the

symplectic quotient In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the ac ...

M

G

at a regular central value

p \in Z(^*)

\mu

. It is defined as the map of equivariant cohomology induced by the inclusion

\mu^(p) \hookrightarrow M

followed by the canonical isomorphism

H_G^*(\mu^(p)) = H^*(M /\!/_p G)

. A theorem of Kirwan says that if

M

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

, then the map is surjective in rational coefficients. The analogous result holds between the

K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, ...

of the symplectic quotient and the equivariant topological K-theory of

M

References

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